Fuzzy integer transportation problem

ZZ'Y sets and systems ELSEVIER Fuzzy Sets and Systems 98 (1998) 291-298 Fuzzy integer transportation problem Stefan Chanas*, Dorota Kuchta Institute of Industrial Engineering and Management, Technical University of Wroctaw, ul. Smoluchowskiego 25, 50372 Wroetaw,Poland Received January 1996; revised November 1996 Abstract An algorithm has been proposed which solves the transportation problem with fuzzy supply and demand values and the integrality condition imposed on the solution. This algorithm is exact and computationally effective,although the problem is formulated in the general way, i.e. its fuzzy supply and demand values can differ from each other and be fuzzy numbers of any type. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy programming; Integer programming; Transportation problem I. Introduction The transportation model has wide practical applications, not only in transportation problems per se, but also in such problems as production planning. The parameters of each transportation problem are unit costs (profits) and demand and supply (production, storage capacity) values. In practice, these parameters are not always exactly known and stable. This paper deals with the case when the unit costs (profits) are known exactly, but the estimate of the supply and demand (capacities) values are only imprecise. This imprecision may follow from the lack of exact information but may also be the consequence of a certain flexibility the given enterprise has in planning its capacities. A frequently used means to express the imprecision are fuzzy numbers and is the approach we have adopted. *Corresponding author. 0165-0114/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved Pll S01 6 5 - 0 1 1 4 ( 9 6 ) 0 0 3 8 0 - 6 In the classical transportation problem with integer demands and supply values there is always an integer solution. This solution can be determined with the transportation simplex method, one of the most popular solution methods of the transportation problem. This property (the possibility of finding an integer solution) is not preserved in the fuzzy transportation problem with fuzzy demands and supplies, even if the characteristics of the fuzzy numbers occurring in the problem are integer. In order to obtain an integer optimal solution (which might be necessary for reasons of feasibility), a special algorithm has to be used. Such an algorithm has been presented in [4]. However, it requires solving a parametric transportation problem with a parameter in the demand and supply values. In this paper we propose an algorithm determining the optimal integer solution of a more general fuzzy transportation problem than the one considered in [-4] making use only of the classical (i.e. non-parametric) transportation problem. 292 S. Chanas, D. Kuchta / Fuzzy Sets and Systems 98 (1998) 291-298 where At and Bj are fuzzy numbers of the following form: 2. Formulation of the problem and notation All fuzzy numbers we will consider in this paper are fuzzy numbers of the L-R type. A fuzzy number A of the L-R type is denoted as A = ( a , d , c~A,flA)L-R and has the following membership function [6]: Ai = (if.z, -ai, o~A,, flA,)L,- R,, Bj = (b j, -~j, ~B~, flB~)~- T~, i = 1,...,m, for t~<a \ #A(t) = ~A /t for t E ~ , d ] 1 (t e R), for t>~d where a, d, CA, fla are real and non-negative parameters and L, R are shape functions. F is a shape function if it fulfills the following conditions: F is a continuous non-increasing function on the half-line [0, ~), F(0) = 1 and F is strictly decreasing on this part of the domain on which it is positive. The following functions are examples of shape functions: • Linear: F(y) = max{0, 1 - y}, y e R + w { O } . • Exponential: F(y) = e -pr, p ~> 1, y e R + w { O } . • Power: F(y) = max{0, 1 - yP}, p >~ 1, y e R + w {0}. • Rational: F(y) = 1/(1 + yP), p/> 1, y ~ R + u { 0 } . Some special cases are possible, in which the type of one or both of the functions L and R may have no significance: • _a=-oo:#A(t)=l for t~<~. • -a= + o O : # A ( t ) = 1 f o r t ~ > a . • C~A = O: lla(t) = 0 for t ~< a. j=l,...,n. The symbol x is a solution matrix. Its elements are corresponding decision variables, i.e. x = [Xu]m× .. The unit transportation costs c~j, i = 1 , . . . , m , j = 1 , . . . , n , are assumed to be crisp numbers. The above formulation of the objective means that the goal is also expressed by a fuzzy number. This fuzzy number is denoted by G and takes the following form: G = ( - ~ , Co, 0,/~G)Lo- Ro. In this place we can say more precisely in which sense the integer fuzzy transportation problem considered in this paper is more general than the one considered in [4]: here the couples (Li, Ri) (i = 1,..., m), (S j, Tj) ( j = 1 .... , n) can all be different, whereas in [4] they have to be identical (linear). In the case when the shape functions are not linear but the same for all parameters, i.e. (L, R~)= (S j, T j) = (F, F), the problem may be easily transformed to the linear parametric transportation problem (see [2, 3]) and solved again by a method proposed in [4]. The following definition makes it clear how the satisfaction of the fuzzy constraints and of the fuzzy goal is understood in problem (1). • flA=O:#A(t)=lfort~>~. The fuzzy transportation problem considered in this paper is formulated as follows: c(x)= ~ ~ cijxij--*n~n, #c(x) = m i n { # A , ~ = l x u ) i = 1 j=l ~ xij~-Ai, (i = l ' ' ' ' ' m ) ' i = 1, ... ,m , j=l ~ xij'~Bj, Definition 1. Let x be an arbitrary solution. (a) The value (1) j = 1, ... ,n i=1 xij >~ 0 and integer, i = 1, ..., m, j = 1, ... , n , is called the degree of satisfaction of the constraints of problem (1) through x. S. Chanas, D. Kuchta / Fuzzy Sets and Systems 98 H998) 291-298 (b) The value 293 A is not less than 2, i.e. A ~ = {t~gl~A(t) ~G(x) = ~ d c ( x ) ) = ~ c~x~j i j=l is called the degree of satisfaction of the goal of problem (1) through x. According to Bellmann-Zadeh approach (see [1]) the optimal solution of problem (1) (called the maximizing solution) is such a solution of the problem which simultaneously fulfills the constraints and the goal to a maximal degree. t> ~}. It is easy to notice that under the assumptions on A~ and Bj accepted in this paper h-cuts A~ and B} are intervals of the following form: A~ = [a__i--Li-l(,~)~A,,ai + Ri-l(2)fla,], i = 1 ..... m, B~ = [bi - Sfl(2)a~j, bj + Tfl()OflBj], j = 1, ... ,n. (3) The 2-cut for the fuzzy goal G is the set G ~ = ( - oc, Co + Rc, l(~)flG]. Definition 2. The maximizing solution of problem (1) is such x for which the function # o ( x ) = rain {~c(X), #G(x)} attains the maximal value. If this maximal value is zero, we say that problem (1) is infeasible. (4) Then we can rewrite problem (2) in the following way: ~ max, c(x) e G ~ , ~ xijeA~, 3. Solution of the problem i= l,...,m j=l According to Definition 2, solving problem (1) is equivalent to solving the following integer mathematical programming problem: xijeB~., j = 1,... ,n (5) i=1 2>0, min{#c(X), #dx)} --. max, xij>lO and integer, i = l .... ,m, j = l , . . . , n . xij >~ 0 and integer, i-- 1,...,m, j = 1 , . . . , n . 2>0, The above problem is not a transportation problem - because of its objective function and the first constraint. For this reason we could not make use of any transportation algorithm to solve it. But we can associate with it an interval transportation problem. This auxiliary problem can be solved by means of any algorithm solving the classical transportation problem (the transition from the interval transportation problem to the classical one will be described in Section 4). Moreover, the solution of this auxiliary problem will permit us to find the solution of problem (5) and thus (1). This auxiliary problem will have the following form (it is defined for any fixed 2 > 0): x~j ~> 0 and integer, i = 1 . . . . . m, j = 1 . . . . . n. c(x) --, min, Solving the above problem is in turn equivalent to solving the following one: 2 --, max, ~dC(X)) >. ;~ , J2A, Xij >~ 2, i= l,...,m, ",.j= 1 #sj xi~ ) 2 , j = 1 . . . . . n, (2) i Let us recall the following well-known definition: Definition 3. Let A be any fuzzy number. The 2-cut of A, denoted by A z, is the set of those real numbers, for which the value of the membership function of ~ xijeA~, i = l . . . . . m, ~ xijeB}, j = 1 ..... n, j=l i=1 xlj/> 0 and integer, i = 1, ..., m, j = 1 . . . . . n. (6) S. Chanas, D. Kuchta / Fuzzy Sets and Systems 98 (1998) 291-298 294 Solving problem (6) for any fixed 2 > 0 allows us to find out whether problem (5) is feasible for this value of 2. It is enough to check whether the objective function value of problem (6) fulfills the first constraint of problem (5). What we need is the maximal value of 2 for which problem (5) is feasible and the corresponding solution of problem (5). The ends of the intervals from the constraints of problem (6) may be non-integer. This would mean that after the transition to the classical transportation problem described in section (6) we might get a classical transportation problem with non-integer supply and/or demand values and the classical algorithm would not guarantee that an integer solution can be obtained. However, we can replace problem (6) with problem (7), having already integer ends in the supply and demand intervals, without changing the set of feasible and optimal solutions. In the definition of problem (7) we use the following notation. Definition 4. Let A be an arbitrary interval. The symbol [A] denotes the widest interval having integer ends and contained in A, i.e. [A] = [a, b], where a = min{t It ~ A, t integer}, b = max{tit e A, t integer}. The problem with which we can replace problem (6) has the following form: c(x) --, min, ~ xije[A~], i = l ..... m, j=l ~'. xi~[B}], j = 1,...,n, (7) i=1 xij >~ 0 and integer i = 1,...,m, j = 1,...,n. The sets of feasible and optimal solutions of problems (6) and (7) are identical - because of the integrality condition imposed on x. For any fixed 2 > 0 we can solve problem (5) converting it into a classical transportation problem with integer supply and demand values (according to Section 4) and applying, for example, the simplex transportation algorithm. If we were able to solve problem (7) (or the corresponding classical transportation problem) as a parametric problem with the parameter 2, we would be able to solve the original problem (1), too. However, the coefficients of the problem depend on parameter 2 in a non-linear way, which makes the matter rather difficult. In order to avoid the necessity of solving a parametric transportation problem, we propose the following algorithm, which only requires solving several classical transportation problems. This algorithm starts from the utmost values of the parameter 2, i.e. 2 = 0 and 2 = 1. We check for which of these parameter values problem (5) is feasible. If it is infeasible for 2 = 0, problem (1) is infeasible. If problem (5) is feasible for 2 = 1, then 2 = 1 is the optimal objective function value in problem (5) and the corresponding solution of problem (5) is the solution of problem (1). If problem (5) turns out to be feasible for 2 = 0 and infeasible for 2 = 1 (which will be the most frequent case), we consider the value 2 = 1 and then the interval (0, ½] (if problem (5) is infeasible for 2 = ½) or [½, 1] (in the other case). 1 Proceeding in this way, we will approach from both sides the optimal value of the objective function of problem (5). Thus, in each step of the algorithm we will consider an interval [21, 22], such that problem (5) is feasible for 2 = 21 and infeasible for 2 = 21. It would not make any sense to divide this interval further, when problem (7) for 2 = 21 is a minimal extension of problem (7) for 2 = 22, as defined in Definition 5. Definition 5. Problem (7) for 2 = 21 is a minimal extension of problem (7) for 2 = 22 when problem (7) for 2 = 21 is identical to problem (7) for 2 = 2", where 2* = m a x { max [~A,(t), max ~Bj(t)} where t is an integer. From the above paragraph it follows that the corresponding algorithm, in which the consecutive 1In fact, these intervals may be slightly modified. The details are given in the description of the algorithm. S. Chanas, D. Kuchta / Fuzzy Sets and Systems 98 (1998) 291-298 intervals of the values of 2 are divided into two, terminates after a finite number of steps, if we check in each step whether problem (7) for 2 = 21 is a minimal extension of problem (7) for 2 = 22. In fact, it is not necessary to check it from the very beginning. The user can determine the length of the interval [-21, 22], beginning with which this condition should be verified. The algorithm works as follows: Step 1: Set 2(1):= 0, 2(2):= 1. Step 2: Solve problem (7) for 2 = 2(1). If the problem is feasible and c(x()~(1)))• Gm), then go to step 3. Otherwise S T O P - problem (1) is infeasible (#o(x) = 0 for all x). Step 3: Solve problem (7) for 2 = 2(2). If the problem is feasible and c(x(2(2)))•G ~2), then S T O P - x(2(2)) is the optimal solution of problem (1) and #o(x(2(2))) = 1. Otherwise go to step 4. Step 4: Set 2(ha/f):= (2(1)+ 2(2))/2 and go to step 5. Step 5: Solve problem (7) for 2 = 2(half). If the problem is infeasible, then set 2(2):= 2(halJ) and go to step 6. Otherwise three cases are possible. (i) #o(x(2(ha/f)))= ~(x(2(ha/f))); then x(2(ha/f)) is an optimal solution of problem (1). STOP. (ii) /ao(x(2(ha/f))) > ~(x(2(ha/f))); then set 2(1):= #c(X()~(half))) and go to step 6. (iii) #o(x(2(half))) < #c(X(2(half))); then set 2(2):= #c(X(2(half))) or, if 2(2) = #c(X(2(half))), then 2(2):= 2(ha/f). Go to step 6. Step 6: If 4 ( 2 ) - 4 ( 1 ) > e , then go to step 4. Otherwise check whether problem (7) for 2 = 4(1) is a minimal extension of problem (7) for 2 = 2(2). If not, then go to step 4. Otherwise S T O P - one of the solutions, x(2(1)) or x(2(2)), is the optimal solution of problem (1). If problem (5) was infeasible for 2 = 2(2), then x(2(1)) is an optimal solution of problem (1). Number e is given by the user. It seems that it should not be greater than 0.1 and not smaller than 0.05. 4. Transformation of an interval transportation problem into a classical one Let us consider the following transportation problem with interval supply and demand values: clxl = 295 i c,jx, min, i=1 j = l ~, xij e [a~,a{], i = 1 . . . . . m, j=l xij • [bJ, b2], j = 1,...,n, i=1 xii~>0andinteger, i = l , . . - , m , j=l,...,n. (8) Problem (8) can be transformed into a classical transportation problem, i.e. with real supply and demand values, by adding origins and destinations with suitable supply and demand levels [4, 3]. The corresponding classical transportation problem is defined in the following way. It will have 2m + 1 origins with the following supply values al, i = 1, ... ,2m + 1: ai=a I fori=l ai = a~-m - al-,n . . . . . m, fori=m+l,...,2m, azm+l = ~ ( b E - b)). j=l It will have 2n + 1 destinations with the following demand values: bj=bJ forj=l bj=b2_,-bJ_, i=1 . . . . . n, forj=n+l,...,2n, j=l The cost coefficients d~j will be defined in the following way: dij = Cij for i = 1 . . . . . m, j = 1 .... , n, d~j = Ci-z,j for i = m + l , . . . ,2m, j = l . . . . . n, dij=ci,j-, fori=l,...,m,j=n+l . . . . . 2n, dgj=C~_m,j_, f o r i = m + l .... ,2m, j=n+ 1, ... ,2n, d~,2, + 1 is a big number, as the corresponding route is prohibited, for i -- 1 . . . . . m, di,zn+l = 0 for i = m + 1, ... ,2m + 1, d2m+ 1,j is a big number, as the corresponding route is prohibited, for j = 1 .... , n, d2,,+lj=0 forj=n+ 1,...,2n. [Xij](2ra+l)x(2n+l) Once we have a solution 2 of the auxiliary, classical problem defined above, = 296 S. Chanas, D. Kuchta / Fuzzy Sets and Systems 98 (1998) 291-298 we can obtain solution of p r o b l e m (8) in the following way: F o r i:= 1 to m do F o r j : = 1 to 2n do xij:= ~ij + ~,,+i,j F o r i:= 1 to m do F o r j : = 1 to n do xi3:= Rij + Ri.,+j Step 1: 2(1):= 0, 2(2):= 1. Step 2: P r o b l e m (7) is feasible for 2 = 0 and c(x(O)) = 150 ~ G O = [0, 800]. Step 3: P r o b l e m (7) is infeasible for 2 = 1. Step 4: 2(ha/f):= (0 + 1)/2 = 0.5. Step 5: P r o b l e m (7) is feasible for 2 = 0.5 and #~(x(0.5)) = 0.74 > pc(X(0.5)) = 0.5. Therefore 5. Computational e x a m p l e The algorithm will be illustrated at the following example: 10Xl~ + 20x12 + 30x13 + 20x2~ + 50x22 + 60X23 --* man, x t l + x12 + x13 ~ (10, 10,5, 5)L-E, XZl + X22 + X23 -- (16, 16, 5, 5)E-L, X l l + X21 ~_ (10, 10, 5, 5)v-E, (9, 9, 4, 4)L-g, ~--- (1, 1, 1, 1)p_R, X12 "t- X22 ~ X13 + X23 xij>lO and integer i=1,2, j=1,2,3. T h e fuzzy goal is determined by the following fuzzy number: G -- (0, 300, 0, 500)L-L. L stands for the linear shape function, E for the exponential with the p a r a m e t e r p = 1, P for the p o w e r shape function with the p a r a m e t e r p = 2 and R for the rational one with p - 1. T h e 2-cuts of the fuzzy supply and d e m a n d values and of the fuzzy goal, with the shape functions a d o p t e d in the example, are as follows (see (3) and (4)): A~ = [ 1 0 - - 5(1 -- 2), 1 0 - - 51n(2)], AZz = [16 + 51n(2), 16 + 5(1 - 2)], B~ : [10 - 5 ~ / O - 2), 10 - 5 ln(2)l, B~ = [9 -- 4(1 -- 2), 9 - 41n(2),l, [1 ~ l = [ 0 , 3 0 0 + (1 - , ~ ) 5 0 0 ] . T h e steps of the algorithm for the example are as follows: ).(1):= Step Step Step 0.5. 6: 2(2) - 2(1) = 1 - 0.5 > 0.05. 4: 2(ha/f):= (0.5 + 1)/2 = 0.75. 5: P r o b l e m (7) is infeasible for 2 = 0.75. Therefore 2(2):= 0.75. Step 6: 2(2) - 2(1) = 0.75 - 0.5 > 0.05. Step 4: 2(ha/f):= (0.5 + 0.75)/2 = 0.625. Step 5: P r o b l e m (7) is feasible for 2 = 0.625 and #~(x(0.625)) = 0.54 < #c(X(0.625)) = 0.64. Therefore 2(2):= 0.64. Step 6: 2(2) - 2(1) = 0.64 - 0.5 > 0.05. Step 4: 2(ha/f):= (0.5 + 0.64)/2 = 0.57. Step 5: P r o b l e m (7) is feasible for ,t = 0.57 and pG(x(0.57)) = 0.58 < #c(X(0.57)) = 0.6. Therefore 2(2) := 0.6. Step 6: 2(2) - 2(1) = 0.6 - 0.5 > 0.05. Step 4: 2(ha/f):= (0.5 + 0.6)/2 = 0.55. Step 5: P r o b l e m (7) is feasible for 2 = 0.55 and #~(x(0.55)) = 0.58 < #c(X(0.55)) = 0.6. As 2(2) = #c(X(0.55)) = 0.6 therefore 2(2):= 0.55. Step 6: 2(2) - 2(1) = 0.55 - 0.5 ~< 0.05. But problem (7) for 2 = 2(1) = 0.5 is not a minimal extension of p r o b l e m (7) for 2 = 2(2):= 0.55 and step 4 is performed once more. Step 4: 2(ha/f):= (0.5 + 0.55)/2 = 0.525. Step 5: P r o b l e m (7) is feasible for 2 = 0.525 and #6(x(0.525)) = 0.7 >/~c(X(0.525)) = 0.5488. Therefore 2(1) := 0.5488. Step 6: 2(2) - 2(1) = 0.55 - 0.5488 ~< 0.05. Therefore it is checked whether problem (7) for 2 = 2(1):= 0.5488 is a minimal extension of p r o b l e m (7) for 2 = 2(1):= 0.55. Since the answer is positive, it is the end of the algorithm. O n e of solutions x(0.5488) and x(0.55) is the optimal solution of p r o b l e m (1). As /~D(x(0.55)) = 0.58 > #o(x(0.5488)) = 0.5488, the second solution, whose full form is given in Table 1, is better. Total cost = 510. #c(X) = 0.6; #~(x) = 0.58; #o(x) = 0.58. S. Chanas, D. Kuchta / Fuzzy Sets and Systems 98 (1998) 291-298 Table 1 Receivers Suppliers 1 2 1 2 3 12 8 1 1 In order to illustrate fully step 5 of the algorithm, we will show how problem (5) is built up and solved for a chosen value of the parameter. Let 2 = 0.55. In this case, A °55 = [7.55, 12.99], A °55 = [13.01, 18.251, B °55 = [6.46, 12.991, B °Ss = [7.2, 11.391, [A °55] = [14, 18], [B°'551 = [7, 12], [B°'551 = [8, 11], a transportation problem with prohibited routes, presented in Table 2. The optimal solution of the above transportation problem is as given in Table 3. The optimal solution of this problem is reduced, after the application of the formulae at the end of Section 4, to the solution of the initial problem (7). This solution is given in Table 1. During the last realization of step 6, while checking the stop condition, 2", defined in Definition 5, is calculated from the following expression: 2" = max #A,(7), #A1(13), /tA2(13), #a2(19),#Bl(6), } = 0.5488 #8,(13),#n2(7 ), ~B2(12), #8~(0), #B3(2) B °'55 = [0.33, 1.82], [A °55] = [8, 121, 297 Problem (7) for 2 = 2* is obviously the same one as for ~,(1), which means a stop of the algorithm. 6. Conclusions [B°S' 1 -- [1, 1]. The transportation problem (5) with interval supply and demand values can be reduced, according to the procedure presented in section 4, to In this paper we have proposed an algorithm solving the integer fuzzy transportation problem (with fuzzy supply and demand values as well as Table 2 1 2 3 4 5 Demand 1 2 3 4 5 6 7 Supply 10 20 10 20 20 50 20 50 30 60 30 60 8 • • 8 1 30 60 30 60 0 0 • 0 0 0 14 14 4 4 8 7 20 50 20 50 0 3 • • 10 20 10 20 0 5 3 4 5 6 7 Supply 1 5 Table 3 1 1 2 3 4 5 Demand 2 7 7 7 1 8 1 1 5 2 3 8 0 4 4 6 14 14 4 4 8 298 S. Chanas, D. Kuchta / Fuzzy Sets and Systems 98 (1998) 291-298 with the fuzzy goal) in the sense of maximizing the joint satisfaction of the goal and of the constraints. This algorithm requires, apart from some easy transformations, only solving the classical transportation problem (in particular, it is not necessary to solve any parametric problems). Fuzzy numbers defining the problem do not have to be trapezoidal. They can differ from each other and be of any type. References [1"] R.R. Bellman and L.A. Zadeh, Decision-making in a fuzzy environment, Management Sci. B17(1970) 203-218. [2] S. Chanas, Parametric techniques in fuzzy linear programming problems, in: J.L Verdegay and M. Delgado, Eds., The Interface between Artificial Intelligence and Operations Research in Fuzzy Environment (Verlag TOV Rheinland, Krln, 1989) 105-116. I3] S. Chanas, M. Delgado, J.L Verdegay and M.A. Vila, Interval and fuzzy extensions of classical transportation problems, Transportation Planning Technol. 17(1993) 203-218. 1'4] S. Chanas, W. Kotodziejczyk and A. Machaj, A fuzzy approach to the transportation problem, Fuzzy Sets and Systems 13(1984) 211-221. [5] R. Dombrowski, Zagadnienia transportowe z parametrycznymi ograniczeniami, Przeglq.d Statystyczny, XV (1968) 103-117. [6] D. Dubois and H. Prade, Operations on fuzzy numbers, Internat. J. Systems Sci. 6(1978) 613-626.